Question

Даны векторы: а{2;-1;0} b{2;3;1} вычислить: ab; b^2 ; (a+b)^2 ; (2a-b)(a+2b)

Answer 1

Надіюся, що ви усе зрозумієте з мого писання)

Answer 2

To compute the given expressions for vectors a and b, we'll first define some vector operations:

1. Dot product (ab): The dot product of two vectors a and b of the same dimension is calculated as follows: ab = a₁ * b₁ + a₂ * b₂ + a₃ * b₃

2. Square of a vector (b^2): The square of a vector b is simply the element-wise square of its components: b^2 = {b₁^2; b₂^2; b₃^2}

3. Sum of two vectors ((a+b)^2): The sum of two vectors a and b can be calculated by adding their corresponding components: (a + b)^2 = {(a₁ + b₁)^2; (a₂ + b₂)^2; (a₃ + b₃)^2}

4. Cross product ((2a - b)(a + 2b)): The cross product of two vectors is defined as follows: (2a - b)(a + 2b) = 2a * a + 2a * 2b - b * a - b * 2b

Now, let's compute each expression:

Given vectors: a = {2; -1; 0} b = {2; 3; 1}

1. Dot product (ab): ab = a₁ * b₁ + a₂ * b₂ + a₃ * b₃ = 2 * 2 + (-1) * 3 + 0 * 1 = 4 - 3 + 0 = 1

2. Square of vector b (b^2): b^2 = {b₁^2; b₂^2; b₃^2} = {2^2; 3^2; 1^2} = {4; 9; 1}

3. Sum of two vectors ((a+b)^2): (a + b)^2 = {(a₁ + b₁)^2; (a₂ + b₂)^2; (a₃ + b₃)^2} = {(2 + 2)^2; (-1 + 3)^2; (0 + 1)^2} = {4^2; 2^2; 1^2} = {16; 4; 1}

4. Cross product ((2a - b)(a + 2b)): (2a - b)(a + 2b) = 2a * a + 2a * 2b - b * a - b * 2b = 2a * a + 2a * 2b - b * a - 2b * b = 2a^2 + 4ab - ba - 2b^2 = 2 * {2; -1; 0}^2 + 4 * {2; -1; 0} * {2; 3; 1} - {2; 3; 1} * {2; -1; 0} - 2 * {2; 3; 1}^2 = 2 * {4; 1; 0} + 4 * {4; -3; 0} - {4; -3; 0} - 2 * {4; 9; 1} = {8; 2; 0} + {16; -12; 0} - {4; -3; 0} - {8; 18; 2} = {8 + 16 - 4 - 8; 2 - 12 + 3 - 18; 0 - 0 - 0 - 2} = {12; -25; -2}

So, the results are: ab = 1 b^2 = {4; 9; 1} (a + b)^2 = {16; 4; 1} (2a - b)(a + 2b) = {12; -25; -2}